Answer: (iii) Polynomial-time reduction from a known NP-complete problem

Explanation:

To prove $L$ is NP-complete you must show:

1. $L \in NP$ (verifiable in polytime), and
2. $L$ is NP-hard: pick a known NP-complete problem $L_0$ (e.g., 3-SAT) and give a polynomial-time reduction $L_0 \le_p L$.

By transitivity of $\le_p$, hardness of $L_0$ transfers to $L$. Subsequent proofs often use Cook's theorem (SAT is NP-complete) as base.

Divide-and-conquer / DP / space calculation do not establish NP-hardness.

Correct choice: (iii)